Rw. Smyth et T. Weinstein, CONFORMALLY HOMEOMORPHIC LORENTZ SURFACES NEED NOT BE CONFORMALLY DIFFEOMORPHIC, Proceedings of the American Mathematical Society, 123(11), 1995, pp. 3499-3506
A Lorentz surface L is an ordered pair (S, [h]) where S is an oriented
C-infinity 2-manifold and [h] the set of all C-infinity metrics confo
rmally equivalent to a fixed C-infinity Lorentzian metric h on S. (Thu
s Lorentz surfaces are the indefinite metric analogs of Riemann surfac
es.) This paper describes subsets of the Minkowski 2-plane which are c
onformally homeomorphic, but not even C-1 conformally diffeomorphic. I
t also describes subsets of the Minkowski 2-plane which are C-j but no
t C-j+1 conformally diffeomorphic for any fixed j = 1, 2,... . Finally
, the paper describes a Lorentz surface conformally homeomorphic to a
subset of the Minkowski 2-plane, but not C-1 conformally diffeomorphic
to any subset of the Minkowski 2-plane.